Since the seminal paper of [1]}, the class \(\cap \)  has been at the frontier of research on total search problems (e.g. [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}).
This research direction is particularly exciting because these problems lie quite low in the  hierarchy, so a-priori any of them could plausibly admit a polynomial time algorithm.
Several natural problems have been identified, and a few complexity sub-classes have been defined (including Very recent breakthrough shows that in fact \(= \cap \)  [10]}!, , , and ).
Yet, until this paper, neither \(\cap \)  nor any of its sub-classes were known to have any naturalIn the sense that all known -complete [2]}, [4]}, -complete [5]}, and \(\cap \) -complete [10]} problems are described in terms of an oracle function succinctly represented by a circuit.
From an algorithm-designer point of view, it is typically more natural -and often much easier- to study the query complexity of such oracle problems.
(The computational complexity of oracle problems is still very interesting, e.g. as a stepping stone for understanding natural problems!)
See further discussions in [16]}, [17]}.
complete problems.
